Dear Marina,

Thank you very much for the nice and detailed comments to my essay. If you liked mine half as much as I liked yours, that's fine! About the analogy with fly/dragonfly, I wanted to make it by using some shape-shifting entity, like larva/butterfly, but larvae couldn't fly! And most important, I didn't want it to be interpreted as if the shape-shift took place in contact with the spiderweb. Regarding the quote you mention, about Wheeler and his students being responsible for many preposterous features of our universe, I am not sure in what proportion it was a joke, or a serious statement, or something between: a metaphor describing the paradigm shifts initiated by them.

You said "I do not quite share your idea that our universe is mathematical." That's fine, since, for example, I don't know if it matters whether a line is a line, or just a collection of points which happen to form a line! Important to me is that I expect an element of the collection to be on that line.

Thanks again for re-reading my essay, and for delighting us with your beautiful one (which I hope it will go into the finals).

Best regards,

Cristi

My congratulations Cristi.

I'm glad you made the finals, and I think your essay will get high grades from the expert reviewers (after all, they are experts). I wish you luck but I don't think you'll need it, as you did an excellent job on your essay this year.

All the Best,

Jonathan

Dear Jonathan,

Thank you, and I am happy for you too! There are probably almost fifty finalists, (and many others interesting essays which are not in the finals), but I hope the expert reviewers will find the time to give them all the deserved attention. I wish you success with your essay!

Best regards,

Cristi

Thank you Cristi for your sweet reply!

And congrats on making the list. For my part, I'm there not on the merit of my entry but because the 'thumbs' were bombing their competition and so pushed many of the best essays bellow the cutoff point, while mine innocuous entry remained under their radar. I already asked for permission to swap my place with one of those unfairly slighted essays.

I returned here to further discuss your fly/dragonfly analogy. It is very clear to me now that we do not understand light. Neither do we interpret it right => hence paradoxes. Regarding my slight change of your analogy, won't you agree that with a single even if weird insect -- rather than 2 types of insects -- there is no paradoxes in either a plain double-slit nor a delayed choice experiment. I.e. we are observing the same phenomenon through different means and so we see its different aspects. But this means that nothing changes retroactively. Do you agree?

Again, congratulations for making the list. Your essay is a good read for SA. Let's hope it gets there,

-Marina

PS Oh! and regarding your take on Wheeler's PAP -- every joke has some truth to it. I've seen people taking his PAP very seriously. I also see PhD's taking Copenhagen interpretations very seriously (as in 'this is IT' rather than just 'Bit' based on lack of better info). So, on the last day of the contest, having read so much that day, I could not longer be sure how you meant it. I thought you were joking when I read it the first time but must have lost my sense of humor by the time I read it again :)

Dear Marina,

Thank you for visiting my page even after the competition is ended, this is very kind. But of course nothing can be kinder than your proposal to swap your place with a less fortunate. Let me tell you that had you withdraw your essay after the votes ended, but before deciding the finalists, the finalists would have been the same (except yours). Because the tie at 4.3, the number of finalists exceeds 40. So I don't think that if you withdraw now, the position will be filled by the next essay, but you can try. Personally, I don't want to check now what is the next essay, but I would like yours to stay in the competition. And I think you deserved the good rates. See that only one FQXi member had to be added in the finalist's list, this shows that the community appreciated good essays, despite some attempts to self-promote by "the representants of the next revolution in physics", by downvoting "the defenders of the evil establishment".

About complementarity. If you allow the particle to morph just before being detected, according to the property you measure, this doesn't quite resolve the weirdness. In the delayed choice experiment with the mirrors, one possibility is that the photon traveled both ways, and the other that it traveled along one of the paths. If we allow it to morph, then this means that it can teleport from one path to the other. So then, we have to admit teleportation of this sort (which is not the same as what goes by the name of "quantum teleportation"). For the sake of the argument, say we admit this too. Then, another problem appear. Let's think at the first mirror, the one doing the splitting. If the photon passes through, the mirror is not kicked. If it is reflected 100%, the mirror is kicked. If it is reflected 50%, the mirror is half-kicked. The delayed choice of whether to keep or remove the second beam splitter affects the existence of the kick in the first mirror. But the kick/not kick already hapened. So here is the mystery, that can't be removed just by allowing instant morphing. Also, how would the instant morphing explain EPR, without assumning the "spooky action at a distance"? All these are predicted by Schrodinger's equation, the mystery appears when we try to tell the story in a local manner.

Best regards,

Cristi

5 days later

Dear Cristinel,

thank you for replying to my post, even though the competition is over. I appreciate your diplomatic way of putting "the representants of the next revolution in physics" vs "the defenders of the evil establishment." lol I'm afraid I'm not that diplomatic. I got the reply from the organizers that I may not substitute my entry for a 'better qualified essay'. I guessed this would be the answer when it was taking long to arrive. I see now that this could make some people question 'why this particular essay and not some other'. Now I hope that, looking for good entries, the judges would go down the list as far as 4.1.

.

Regarding light, I am afraid what I meant to say was that the whole paradigm with 'particle' vs 'wave' is in deep trouble. Especially the 'particle' part. The reason I liked your quantum spiders analogy so much (with my slight modification) was because it allowed to let go of old analogies and --even if for a moment-- take a fresh look at the old problem. And 'particles' is just another analogy (more on this later).

Here is how Wheeler's delayed choice experiment could be translated into you modified by me model:

Once upon a time, a weird quantum insect flew out of its nest and, having traveled for millions of years, was caught by a very hungry quantum spider, who had not eaten a thing since last week when it hatched. To catch its prey, the spider used a specific net designed to reveal a particular aspect of this weird thing; and the net could make it appear as if it was either a fly or a dragonfly, but, regardless of the type of net the spider used and how the thing turned out, it was still the same insect.

See? There is no paradox.

The paradox comes up when we speak of 'particles', which, to me, indicates that that the model does not quite work here. If we examine the history of how this particle idea came to occupy such a prominent place in quantum theory, ruffly: it began with Newton and, having proved to be very useful in several areas of physics, went through its ups and downs in importance until it was first revived and then promoted to the central role by quantum mechanics.

So, when you say, "one possibility is that the photon traveled both ways, and the other that it traveled along one of the paths" -- what is this description based on? What if entirely something else is taking place that accounts for the observations?

The trouble with 'particles' is that they are dimensionless points in empty space, while the real phenomenon appears to be some dynamic configuration of the medium we call space (in other words, you cannot extract it from the medium it is in). We do not quite understand this dynamic configuration; and this causes us to come up with strange ideas, which, in turn, lead to paradoxes. The famous Feynman's phrase about the quantum weirdness notwithstanding, the paradoxes are the symptom of the model not being right -- as in not right for a the application at hand. And I understand that people get attached to a model -- unduly so, if they have invested a lot of time and effort into it. Still, I don't see how the quantum theory can advance with this old, riddled with paradoxes particle model in place.

In my understanding of these experiments (and I hope it's right, else please correct) photons do not interact with each other, but only with electron clouds of the 'material things' represented by the mirrors, or the screen with slits, or the detection screen, or the array of snooping devices aimed to detect which path a 'particle' took, etc. The important thing here is to remember that observation of a 'particle' constitutes placing some 'material' obstacle on its path, (which, naturally, affects it).

Again, when you wrote, "If you allow the particle to morph just before being detected" -- who said anything about morphing? That too is an assumption presumably based on the Copenhagen interpretation. That too is just another analogy.

I sincerely hope you would answer my questions, because.. ..cause I really would like to know :)

Thank you,

-Marina

Dear Marina,

I fully agree that particles are not points. They are wavefunction, state vectors in the Hilbert space. It may happen that the wavefunction is concentrated at a point, but this can be only for an instant. So the point particle representation wouldn't help here. When we measure some properties, only some of the possible states have that property. So you are right to say that, no matter whether we measure flies or dragonflies, it is the same insect. Because a wavefunction which has the property named position well defined, can morph into one that has the momentum well defined, although these are complementary properties. So, from this viewpoint, I agree that there is one insect, which can turn out to be of one kind or another. On the other hand, there is a reason why I emphasized the distinction between flies and dragonflies: I think that the delayed choice experiment shows that the photon already was either in a "both-ways" or a "which-way" state, when it reached the first half-silvered mirror. If this is not convincing, think at another version, called the quantum delayed choice experiment. In this experiment, we can delay even the decision of whether the second half-silveerd mirror is present, and maintain this choice in a superposition for an undetermined time.

My point of discussing the delayed choice experiment was that its conclusion is, in my opinion, different than Wheeler's, which is that "it" doesn't exist except through the "bit" we observe. My proposal is that the wavefunctions are real (ontological, not only epistemic), unitary evolution describing their evolution is not broken, but the initial conditions are not fixed yet completely (delayed initial conditions). But each measurement fixes them more and more. Nonlocality and the apparent backward causality exhibited by some experiments show in fact that Nature reserves the freedom to choose the initial condition at a later time, and does this partially with each new measurement. While this may seem strange, in my opinion explains the weirdness of quantum mechanics. Moreover, it become less strange, if we think in terms of the 4D block world, as in relativity, and we realize that global consistency principle has to hold.

Now, I think that various interpretations can tell the same story in different ways. The one I propose makes more sense to me, but I would not exclude the possibility that for others, other interpretations make more sense. And maybe one interpretation is more helpful than another, for a particular problem. For instance, I am most interested in having an ontological interpretation which doesn't violate unitary evolution, and in the relation between relativity and the quantum.

Best regards,

Cristi

Thank you Cristi for your elucidating reply!

The arXiv paper you linked makes a wonderful illustration, even though they too say a photon 'morphs' between "particle" and "wave" depending on how we 'look' at it. But why would it morph rather than perhaps being already something in-between or both?

See, I'm trying to look at a photon as a disturbance propagating through space, and, doing so, I look at the experimental area as a whole from the POV of the dynamic structure of space that 'wants to be empty' locally. Then whatever disturbance enters a locality is expelled into the direction that gives. 'Measuring' or 'looking' introduces another disturbance into this setup area, which affects how the preexisting disturbance behaves.

I am only trying to understand this mysterious thing we call space ..and light. I am perplexed trying to explain the most mundane experience of seeing within the current scientific framework. ..and to me it appears that neither 'wave' nor 'particle' are appropriate models for light. Here is why:

When I look at the world, what I see are the photons registered on the retina of my eyes. Back in the XV century Renaissance artists realized that straight streams of light converging on a point of view were responsible for the appearance of the perspective and the illusion of depth on a flat canvas -- it's all pure geometry. But. In reality, the same streams of photons are coming from everywhere, from all directions at once. If photons are particles, why would not they bounce off each other and scatter? And if they are waves, why would not they interfere and thus change their form before reaching my eyes? In other words, how come I get to see such a clear picture? I know that, within the current scientific framework, photons do not interact with each other and that should be enough to 'explain' this. But... somehow this does not explain it, really. Something crucial seems missing -?

In this regard, I was wondering if you knew about the experiments that involve 3 streams of photons along the x, y, z directions, focused at one point, and, after crossing this point, made to go each through their own screen with slits -?

Thank you very much for your reply and I am still trying to understand what exactly you mean by wavefunctions being real ontologically and not just epistemically. The best I can do is to imagine them as underlying quantum processes that govern the dynamic structure of space -?

Thanks a lot,

-Marina

Dear Marina,

We agree in several points. As you observed, photons are not point particles, and they don't bounce, they pass through one another, without interacting, indeed. I am not aware about the experiment you mention, but I think that the three streams of photons don't "see" each other. The fact that photons pass through each other undisturbed doesn't contradict interference, in fact interference wouldn't be possible without this. The problem is that, if after interference, you detect the photons, they are found in positions distributed according to the Born probability obtained from the wavefunction which resulted from interference. It is as if the wave, initially widely spread, concentrates at the point where it is found. As I said earlier, when you measure position, you find the photon localized around a point, but this doesn't mean it is point particle, but just a localized wave. So, the wavefunction before observation is epistemic, it gives the probabilities, and after the measurement, it is ontological, it is what you find. There are more ways to consider the wave to always be ontological. The most direct way is to consider that, just before being measured, it changes its shape, to become an eigenstate of the measured observable. This picture is like this: (1) The wave is ontological. (2) It evolves unitarily between measurements. (3) At measurement it is projected, according to the projection postulate (and Born's probability). This is probably close to your idea of shape shifting. It explains some features, and if this answers all of your questions, it may be what you want. It is not very good when you try to think how this applies in spacetime, especially when entanglement is involved, but it challenges one's intuition even for only one particle (by particle I don't mean point particle). For some, this picture may be sufficient, for me is not. It may be just a matter of preference after all. I explained 5 years ago some of my reasons why I am not happy with it, and what I proposed in place, in this video, and this article. I hope this will help.

Best regards,

Cristi

Dear Cristi,

thank you for your reply! I watched the video and looked at your paper. You have a beautiful voice :) even though some whistles were introduced by the loss of resolution.

I really appreciate your feedback on my thoughts. You wrote: "The fact that photons pass through each other undisturbed doesn't contradict interference, in fact interference wouldn't be possible without this."

But.. in the double-slit experiments, the interference is the result of diffraction of light streaming through the slit -- the photons bounce off the electron clouds (or weak magnetic field) of the screen material. And it seems that a weak polarization also takes place -? That's why a precise setup is required between the wavelength, the width of the slits, the separation between them, and the distance to the detection screen, on which the interference itself is produced.

Artists have long noticed this weak diffraction/polarization of light along the edges of the objects. Being mindful of it is what makes an object in a still life appear voluminous and integrates into its surrounding rather than appearing flat against background.

So, in my understanding, there is no interference between the photons themselves. There is interference between the photons and the weak magnetic fields (electrons) of the material objects. Otherwise, if photons, streaming from all directions, interfered before reaching my eyes, then the forms of their waves would be modified; and a modified light wave implies at least a different color or its intensity -- but nothing of the sort happens, nothing undulates or shimmers, and I see clearly -?

Again, in my understanding, the waves of the same kind and frequencies do interact with each other, don't they? But light in empty space does not seem to (at least, not until it reaches the detection screen).

Thank you very much on your feedback, I value it a lot,

-Marina

Dear Marina,

Yes, in the double slit experiment, diffraction is required. Diffraction is just light moving around obstacles, and this can be understood by Huygens's principle (every point on a wavefront is a source of a new wavelet, and all wavelets combined give the wave), and not by bouncing. According to this principle, wavelets travels along various paths, most of them curved, although when we add them, we obtain that light travels in straight lines. Diffraction appears when in the sum of the wavelets part of them are missing (when obstacles are present), this is why in diffraction light doesn't go straight.

On the other hand, interference is just the superposition (addition) of the waves. When two waves interfere, we add the amplitudes at each point, taking into account the relative phase. This is why I said interference is not interaction. Waves combine their strength at each point, but they don't interact. So, when you say "So, in my understanding, there is no interference between the photons themselves. There is interference between the photons and the weak magnetic fields (electrons) of the material objects", you surely mean "there is no interaction between the photons themselves". Also, you say "Again, in my understanding, the waves of the same kind and frequencies do interact with each other, don't they?". They don't interact. The fringes observed in the experiment are not due to interaction, but to superposing the two waves.

Best regards,

Cristi

5 days later

Thank you for your reply, Cristi!

I did imagine interference to be another kind of interaction. I see now that I was wrong. What I get now is that light waves are akin to ripples on the surface of water, reaching an electron cloud surrounding a nucleus, which is like a cork bobbing on the surface of a pond. This makes up a good 3D model of what may be going on in 4D (3D being the surface of a 4D "object" like a hypersphere). This brings to mind Kaluza (without Klein's unnecessarily compactificataion of the 4th dimension). This looks so natural, with EMR being confined to the 3D surface, exactly like ripples on the surface of a pond; and "gravity" being the curving of this surface, exactly per Einstein. What objections are there to this implicit model of space? Why is not it universally accepted?

Thank you very much for your reply :) I very much value your feedback,

-Marina

Dear Marina,

Kaluza's idea is great. To get the electromagnetic field from the geometry of the fifth dimension of a curved 5-dimensional spacetime, some symmetry conditions have to be imposed to the geometry. The main problem is, what takes care of these conditions to be satisfied everywhere? If the fifth dimension is like the others, it does not necessarily have these symmetries. Also, from experimental viewpoint, there is no way to see the fifth dimension, unless those symmetry conditions are broken. Yet, the very existence of the electromagnetic field can be viewed as proof of the fifth dimension, subject to those conditions. Now, the thing is that, by imposing these conditions, what we obtain is a special case of space, named technically principal bundle, with gauge group (the group of symmetries) given by the rotations in plane. The group of these rotations is, from geometric viewpoint, a circle, so the fifth dimension is compactified, but it can't be probed anyway, not because it is small, but because of the gauge invariance. So, the gauge theory viewpoint, which is the most accepted, is in fact the Kaluza-Klein viewpoint. At least from the viewpoint of geometry. But most physicists see the phase parameter (the fifth coordinate) to be just a parameter, and don't load it with a geometric interpretation. This may be a smart move, because there are other ways to introduce gauge dimensions, than simply adding dimensions. For example, the role of the extra fifth dimension can be played by the phase of the wavefunction of the electron, whatever this is. So, we keep what is good, but try to assume as little as possible.

Best regards,

Cristi

Dear Cristi,

thank you for your reply! You speak of the 5th dimension, while I speak of only 4, all of them spatial though. In this regard, I have a question about the GR:

In GR, the spacetime is said to be 4-dimensional, 3 are spatial and 1 is time. GR is also said to be geometrical. But, in simple examples where geometry is commonly employed, like tracing a trajectory of a cannonball, we always align the time dimension with one of the spatial dimensions, in the direction of the movement.

Imagine a simple graph: a thick horizontal line at the bottom, a cannon 'standing' on this line on the left and a dash-line of the parabola traced by the cannonball in the 'sky'. On this simple 2D X,Y graph, Z spatial dimension is missing; however it is implied. Time dimension is also present: it is simply added, superimposed, with the X spatial dimension. Here the spacetime is 2D (idealized parabola on a plane). Both dimensions are spatial. The time dimension is superimposed on the pre-existing 2D space. And please note that the time dimension is always aligned in the direction of the movement -- imagine it was going perpendicularly, or even in the opposite direction lol -- but why not, if it is simply a dimension just like space?

Now, the Einstein's 4D field equations. Why can't I say that they too are about the 4-dimensional space (as in 4 spatial dimensions) with time being superimposed on this 4-space to show the 'movement'? Of course, the movement here is more complex than in the cannonball example, but it is still a movement in space. Doesn't it show the dynamic curving of a 3D surface of a 4D "object", just like artists commonly illustrate it in popular science magazines?

In other words, please give a good argument that in the GR field equations, the time dimension is not superimposed on 4D space, just like it is in the cannonball example.

Thank you very much for your feedback :)

Best regards,

-Marina

Dear Marina,

Kaluza-Klein theory has 4 space dimensions plus one time dimension. If I understand well, you want to replace time by a space dimension. But the analogy of cannonball really is in 1 space plus 1 time, not just in 1 space. So, if you want to use this analogy and add a fourth space dimension, it seems to me that the fifth is kept, and is time. Unless I don't understand what you mean.

Sometimes is useful to make time to look like space. There is a trick of this type, named Wick rotation, used to solve some problems in quantum field theory and quantum mechanics. Also you may want to look for imaginary time, and Hartle-Hawking state.

But I am sure you don't have this in mind either.

Maybe you will find what you want in this essay, I think you can contact the author on his this year's essay page. He may have some ideas related to what you said, but I don't know how far did he advance with them.

Frankly, I don't think I understand what you mean, and what problems do you think this can solve, to beat the way relativity is done. To know how to continue this discussion, may I ask you how well do you know general relativity? I would ask you to pick one of these:

(1) "I know about special and general relativity for pop-science books or documentaries, with no equations at all. I want to discuss using analogies with school level mechanics"

(2) "I understand special relativity including the equations, and also electrodynamics. I know general relativity only at pop-sci level."

(3) "I understand both special and general relativity including with equations, and differential geometry as it is used in general relativity."

(4) "(3), but in addition, I am familiar at the mathematical level with Kaluza-Klein and gauge theories."

Then, if your choice is greater than (1) you may try to translate your arguments from (1) to the level of your choice. Maybe you can compare your ideas with those from GR, and show what would be the advantages of your theory. You asked me for a good argument for GR against your proposal, but I don't see what are the arguments supporting your proposal. GR has plenty of arguments supporting it.

I want to confess that I don't find very productive the discussions about physics which remain at level (1). This may help understanding, build an intuition, but it is just a first step. More efficient arguments can be made at the more advanced levels.

Best regards,

Cristi

Dear Cristi,

thank you for your reply :) To answer your questionnaire, I am definitely above the first level, without the technicalities of course, which probably would place me at level 2 (-? I was not quizzed in school on this).

But here is what I meant with my question. I am not looking for ways to "to beat the way relativity is done". I am only looking for the evidence that the universe is actually 4-dimensional (as in 4 spatial dimensions) and this requires not invention of something new but a slightly different way of looking at the already familiar.

If you consider the historical context in which relativity and Minkowski spacetime appeared, the 4th spatial dimension was the hottest topic in both popular culture and science at the time (more so in the popular culture, a 'little' fact largely forgotten today). The relevant science was vigorously developed during the XIX century, and in the last quarter of it, the ideas spilled from the universities into the popular culture and captured the people's imagination.

What followed was that already by the turn of the XX century, the popular culture took over the subject of the 4th dimension to such an extent that it became unseemly for a serious scientist to even mention it.

Just to give you an idea of the atmosphere that surrounded the topic at the time, spiritualists were the first to "colonize" the 4th dimension. It became the "afterlife" where spirits roamed and from where they "communicated" with the living -- them being in the 4th dimension "explained" why they could not be seen directly. But this was just the tip of the iceberg. Several popular books were published, of which Edwin Abbott's Flatland is the most notable and survived to this day. The 4th dimension was mentioned in philosophy and literature, employed in art and even music. The marketplace was abuzz with most wild speculations. Basically, at the time, everything unexplained or simply misunderstood was promptly dispatched into the 4th dimension -- and this was supposed to "explain" it all.

This was the cultural context in which relativity and Minkowski spacetime appeared. To speak of the 4th spatial dimension, other than in the abstract areas pertaining exclusively to geometry and topology, was equivalent to debasing science to the level of the marketplace -- and physics is not an abstract science. Physics is about reality. To speak of the reality of the 4th spatial dimension, at the time, was akin to giving validity to all the craziness that went on. The 4th spatial dimension became almost an unspeakable topic in physics simply due to this cultural pressure.

Now, knowing this, I wonder how physics would develop without this cultural context. My hunch is that Minkowski went out of his way to veil the 4-dimensional setting by stressing that the 4th dimension was time -- after which it became the tradition and the way of looking at things. And so with my question I wanted to point out that, when we speak of movement, in all our models, time is always simply aligned with one of the preexisting spatial dimensions.

In any rate, thank you for allowing me to pick your brilliant brains. I found that I learn the quickest from a few words of a specialist like you than from studying hundreds of pages of books. But I understand your reluctance to participate in this one-way exchange. I only wanted to consider the reality of the extra spatial dimensions -- now that the cultural context is vastly different lol.

Again, thanks a lot :)

Best regards,

-Marina

Dear Marina,

"This was the cultural context in which relativity and Minkowski spacetime appeared."

I would not credit too much the "market" eager to find where to place the spirits, and avid of magical tricks involving the fourth dimension, for the following reasons.

In 1905, Einstein did not mention time as a fourth dimension. He only realized that the Lorentz-Fitzgeranld and Poincare transformations are a good replacement of Galilei's transformations, able to accommodate electrodynamics.

In 1905, Poincare realized that Lorentz transformations are in fact rotations in a four dimensional space, where one of the dimensions was imaginary. Rotations were already known in any number of dimensions. They are the linear transformations which preserve a quadratic form. It was known that not all quadratic forms can be diagonalized to have only 1 on the diagonal, some of them had some -1. This was the case for Lonrentz's transformations.

In 1907, Minkowski developed this idea more, and took it more literally than Poincare.

In my opinion, it was not the cultural context you mention which triggered this, but rather both were possible by the mathematical developments of the XIX century. Spaces with more dimensions were known to mathematicians. Quaternions, which live in 4 dimensions, were also known. An important role was also played by the non-Eucldiean geometries and Riemannian geometry.

I think that the main paradigm which made time in special relativity to be interpreted as a fourth dimension was Klein's Erlangen program. The idea that symmetry groups are at the core of the newly appeared geometries was so simple, and efficient, and the work of Lorentz, Einstein, Poincare, and Minkowski, made it to be applicable to special relativity too. This is why time was accepted as the fourth dimension. Einstein's idea of relativity of simultaneity becomes so natural when we realize that the different observers are oriented along different directions in a four dimensional spacetime. Lorentz transformations are rotations in the fourth dimension, so by changing the velocity, we rotate out of our initial space, in time. Of course, because the quadratic form which is preserved during the Lorentz transformations, is not positive definite, there are directions which cannot be rotated one into the other. Space cannot be rotated to become time and conversely. Space and time directions are separated by the light cone. This prevents us for accelerating until we can travel in our past. So, we may say that we cannot actually test that time is the fourth dimension, more than a parameter, by traveling in time as we travel in space. But except this, which is predicted by the theory itself, the evidence is abundant. Energy and momentum become married in four dimensions, electric and magnetic potentials too, and many other physical quantities are unified like this. By seeing that Poincare invariance is so universal, and that indeed lengths and durations depend on the observer, we realize that all possible evidence is there. Moreover, just by combining Poincare invariance with quantum mechanics, maintaining the spirit of the Erlangen program, we obtain the particles, classified by the spin. The evidence is overwhelming, and it can't be just a package of special relativity, to be sold to a marked intoxicated with ideas about fourth dimension. This evidence tastes better when you try to understand it mathematically. Then, you see that all parts are connected, and not a collection of disparate ideas that can be modified or replaced with other ideas so easily.

Best regards,

Cristi

Dear Cristi,

thank you for your reply :) I'm afraid you completely misunderstood what I was saying about how the marketplace influenced the development of physics in the early XX century. You said "In my opinion, it was not the cultural context ... which triggered this, but rather both were possible by the mathematical developments of the XIX century." Exactly!

My point was that, at the time, the whole society was so intoxicated with the idea of the 4th dimension that already by the turn of the XX century, the mere mention of it provoked in most people a knee-jerk-like reaction of the type 'please! not again!' The 4th dimension had become synonymous with the most wild, unbridled speculation. That's what I meant by the 'cultural context'.

Imagine there was no internet, no TV and not even radio. What most people did for entertainment was talking, speculating about various tings; and the 4th dimension was the main 'entertainment' for decades. A new generation, including those who later on defined the developments in physics, grew up hearing or reading about it almost daily, in various contexts, often one crazier than next.

And so my point was that young physicists in the first quarter of the XX century were negatively conditioned against the 4th dimension. I believe this is why the idea was not pursued as much as it should have been; and why, despite the overwhelming mathematical evidence, the mainstream physics --100 years later!-- still has not accepted its reality (and by extension, society as a whole).

I researched this info couple of years ago, trying to understand why this happened; and my conclusion was that in the beginning of the XX century on, it became almost impossible to present novel ideas that involved the 4th spatial dimension. The cultural setting was such that a mere mention of 4D was enough to make most people wince! At the time, it took courage to seriously speak of the 4th D -- no one wanted to be dismissed outright by mere association with the topic that by then had become synonymous with wild speculations. That's how, I believe, the marketplace influenced the development of physics in the XX century.

As for the history of the 4th dimension in mathematics, maybe you remember the last-year essay by Renate Quehenberger . I thought it contained one of the best historical reviews of the 4th dimension in science 'in 9 pages or less'.

Again, thank you for your feedback, I'm glad we agree that the mathematical evidence for the 4th spatial dimension is overwhelming. Now that the cultural setting is vastly different and it's once again cool to talk about the 4th dimension, let's hope it gets accepted by the mainstream soon :)

Again, thank you for your feedback!

Best regards,

-Marina

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  • Post #260

PS

re mathematical evidence, I wanted to go further than what you said above and so well-reviewed by Renate in her 2012 essay. I wanted to make a case that our beloved general relativity, the jewel in the crown of physics, is in fact set in 4 spatial dimensions. It's a case of one way of looking at things vs another, which is conventional -- but it could have been the other way around. I am reading again Minkowski 1908 talk Space and Time - pages XV on.

..oops! I found this PDF looking for the best original Minkowski but only now, when I checked that the link I gave you worked, I looked at the preface and it's just what I've been looking for!

Thank you again, Cristi! I knew that talking to you would be most productive :)

-Marina

Dear Marina,

Thank you for the clarification. I think you have a very good sense of how the public's expectations were at that time. You say "I'm glad we agree that the mathematical evidence for the 4th spatial dimension is overwhelming". So far, the evidence pointed to a fourth dimension which is time, and the square of a timelike vector has opposite sign than the square of a spacelike vector. In this respect, there is a difference between space and time. So, the arguments I mentioned support the idea of time as the fourth dimension. This point is widely accepted by the mainstream, although they may not all acknowledge it. For example, the mainstream demands theories to be Lorentz invariant, hence they accept the fourth dimension. But many of them don't like the consequence, that space and time form a continuum, a block world, and this is why I said "they accept, but they may not acknowledge". On the other hand, I don't know of arguments supporting the idea of a fourth space dimension, other than those coming from the Kaluza-Klein theory and variations. In "variations" I include superstring theory, and here, again, there is a curious phenomenon. The mainstream is divided. There are so many working in superstring theory, and so many papers, and so many supporters, and there are in the same time many skeptics. But my feeling is that most accept with more ease the extra dimensions in superstring theory, but doubt the possibility that time itself is a dimension as well. Accepting extra dimensions as in string theory requires to accept something which is not directly seen, but seems to explain a lot of what we see. On the other hand, accepting time as a fourth dimension gives us the impression that future is already there, and it is fixed, and we know that we can choose freely. I explained why the two don't necessarily contradict one another in page 5 of my essay, and more in refs. [17-19].

Best regards,

Cristi